Quantum Half-Planes via Deformation Quantization

We demonstrate the main idea of constructing irreducible unitary representations of Lie groups by using Fedosov deformation quantization in the concrete case of the group Aff® of affine transformations of the real straight line. By an exact computation of the star-product and the operator ${\widehat{\mathrm{\ell }}}_Z$, we show that the resulting representations exhausted all the irreducible representations of this groups.